It is commonly assumed, based on hydrodynamic simulations, that the
neutral hydrogen in the inter-galactic medium (IGM) can be related to
the underlying mass distribution. It is then possible to estimate the
matter power spectrum on scales of a few megaparsecs from the absorption
observed in quasar spectra, the so-called Lyman-alpha forest.
The usual procedure is to measure the power spectrum
of the transmitted flux, and then to infer the mass power spectrum.
Photo-ionization heating by the ultraviolet background radiation and
adiabatic cooling by the expansion of the Universe combine to give a
simple power-law relation between the gas temperature and the baryon
density. It also follows that there is a power-law relation between
the optical depth and
b.
Therefore, the observed flux
F = exp(- ) is
strongly correlated with
b,
which itself traces the mass density.
The matter and flux power-spectra can be related by

(1.18)

where b(k) is a bias function which is calibrated from
simulations. Croft et al.
[33]
derived cosmological parameters from Keck Telescope observations
of the Lyman-alpha forest at redshifts z = 2 - 4.
Their derived power spectrum corresponds to that of a CDM model,
which is in good agreement with the 2dF galaxy power spectrum.
A recent study using VLT spectra
[34] agrees with the
flux power spectrum of Ref.
[33].

This method depends on various assumptions. Seljak et al.
[35] pointed out
that errors are sensitive to the range of
cosmological parameters explored in the simulations, and the treatment
of the mean transmitted flux.
Combination of the Lyman-alpha data with WMAP suggested deviation from
the scale-invariant n = 1 power spectrum
[7,
6], but
Seljak et al. [35]
have argued that the combined data set is still compatible with n
= 1 model.